Topics
Units and Measurements
- Quantitative Science
- System of Units
- Derived Quantities and Units
- Rules and Conventions for Writing SI Units and Their Symbols
- Measurement of Length
- Measurement of Mass
- Measurement of Time
- Dimensions and Dimensional Analysis
- Accuracy, Precision and Uncertainty in Measurement
- Errors in Measurements>Systematic Errors
- Errors in Measurements>Random Errors
- Estimation of Errors
- Combination of Errors
- Significant Figures
- Definitions of SI Units and Constants
Mathematical Methods
- Vector Analysis
- Scalar
- Vector
- Vector Operations>Multiplication of a Vector by a Scalar
- Vector Operations>Addition and Subtraction of Vectors
- Vector Operations>Triangle Law for Vector Addition
- Vector Operations>Law of parallelogram of vectors
- Resolution of Vectors
- Multiplication of Vectors
- Scalar Product(Dot Product)
- Vector Product (Cross Product)
- Concept of Calculus
- Differential Calculus
- Integral Calculus
Motion in a Plane
- Concept of Motion
- Rectilinear Motion
- Displacement
- Path Length
- Average Velocity
- Average Speed
- Instantaneous Velocity
- Instantaneous Speed
- Acceleration in Linear Motion
- Relative Velocity
- Motion in Two Dimensions-Motion in a Plane
- Average and Instantaneous Velocities
- Acceleration in a Plane
- Equations of Motion in a Plane with Constant Acceleration
- Relative Velocity in Two Dimensions
- Projectile Motion
- Uniform Circular Motion (UCM)
- Key Parameters of Circular Motion
- Centripetal Acceleration
- Conical Pendulum
Laws of Motion
- Fundamental Principles of Motion and Mechanics
- Types of Motion
- Aristotle’s Fallacy
- Newton’s Laws of Motion
- Newton's First Law of Motion
- Newton’s Second Law of Motion
- Newton's Third Law of Motion
- Inertial and Non-inertial Frames of Reference
- Types of Forces>Fundamental Forces in Nature
- Types of Forces>Contact and Non-Contact Forces
- Types of Forces>Real and Pseudo Forces
- Types of Forces>Conservative and Non-Conservative Forces
- Types of Forces>Work Done by a Variable Force
- Work Energy Theorem
- Principle of Conservation of Linear Momentum
- Collisions
- Elastic and Inelastic Collisions
- Perfectly Inelastic Collision
- Coefficient of Restitution e
- Expressions for Final Velocities in Elastic Head-On Collision
- Loss of Kinetic Energy in Perfectly Inelastic Head-On Collision
- Collision in Two Dimensions
- Impulse of a Force
- Necessity of Defining Impulse
- Rotational Analogue of a Force: Moment of a Force Or Torque
- Couple and Its Torque
- Proof of Independence of the Axis of Rotation
- Mechanical Equilibrium
- States of Equilibrium
- Centre of Mass>Mathematical Understanding of Centre of Mass
- Centre of Mass>Velocity of Centre of Mass
- Centre of Mass>Acceleration of Centre of Mass
- Centre of Mass>Characteristics of Centre of Mass
- Centre of Gravity
Gravitation
- Concept of Gravitation
- Kepler’s Laws
- Law of Orbit or Kepler's First Law
- Law of Areas or Kepler's Second Law
- Law of Periods or Kepler's Third Law
- Newton's Universal Law of Gravitation
- Measurement of the Gravitational Constant (G)
- Acceleration Due to Gravity (Earth’s Gravitational Acceleration)
- Variation in the Acceleration>Variation in Gravity with Altitude
- Variation in the Acceleration>Variation in Gravity with Depth
- Variation in the Acceleration>Variation in Gravity with Latitude and Rotation of the Earth
- Variation in the Acceleration>Effect of the shape of the Earth
- Gravitational Potential Energy
- Expression for Gravitational Potential Energy
- Connection of Potential Energy Formula with mgh
- Potential and Potential Difference
- Escape Velocity
- Earth Satellites
- Projection of Satellite
- Weightlessness in a Satellite
- Time Period of Satellite
- Binding Energy of an Orbiting Satellite
Mechanical Properties of Solids
- Mechanical Properties of Solids
- Elastic Behavior of Solids
- Stress and Strain
- Types of Stress and Corresponding Strain
- Hooke’s Law
- Elastic Modulus>Young’s Modulus
- Elastic Modulus>Bulk Modulus
- Elastic Modulus>Modulus of Rigidity
- Elastic Modulus>Poisson’s Ratio
- Stress-strain Curve
- Strain Energy
- Hardness of Material
- Friction in Solids
- Origin of Friction
- Types of Friction>Static Friction
- Types of Friction>Kinetic Friction
- Types of Friction>Rolling Friction
Thermal Properties of Matter
- Thermal Properties of Matter
- Temperature and Heat
- Measurement of Temperature
- Absolute Zero and Absolute Temperature
- Ideal Gas Equation
- Thermal Expansion
- Linear Expansion
- Areal Expansion
- Volume Expansion
- Relation Between Coefficient of Expansion
- Specific Heat Capacity
- Specific Heat Capacity of Solids and Liquids
- Specific Heat Capacity of Gas
- Heat Equation
- Thermal Capacity
- Calorimetry
- Change of State
- Analysis of Observation>From Point A to B
- Analysis of Observation>From Point B to D
- Temperature Effects and Considerations
- Evaporation vs Boiling
- Boiling Point and Pressure
- Factors Affecting Cooking
- Sublimation
- Phase Diagram
- Gas and Vapour
- Latent Heat
- Heat Transfer
- Conduction
- Thermal Conductivity
- Coefficient of Thermal Conductivity
- Thermal Resistance
- Applications of Thermal conductivity
- Convection
- Application of Convection
- Free and Forced Convection
- Radiation
- Newton’s Law of Cooling
Sound
- Sound Waves
- Common Properties of All Waves
- Transverse Waves
- Longitudinal Waves
- Mathematical Expression of a Wave
- The Speed of Travelling Waves
- The Speed of Transverse Waves
- The Speed of Longitudinal Waves
- Newton's Formula for Velocity of Sound
- Laplace’s Correction
- Factors Affecting Speed of Sound
- Principle of Superposition of Waves
- Echo
- Reverberation
- Acoustics
- Qualities of Sound
- Doppler Effect
- Source Moving and Listener Stationary
- Listener Approaching a Stationary Source with Velocity
- Both Source and Listener are Moving
- Common Properties between Doppler Effect of Sound and Light
- Major Differences between Doppler Effects of Sound and Light
Optics
- Fundamental Concepts of Light
- Nature of Light
- Ray Optics Or Geometrical Optics
- Cartesian Sign Convention
- Reflection>Reflection from a Plane Surface
- Reflection>Reflection from Curved Mirrors
- Total Internal Reflection
- Refraction of Light
- Applications of Total Internal Reflection
- Refraction at a Spherical Surface and Lenses
- Thin Lenses and Their Combination
- Refraction at a Single Spherical Surface
- Lens Makers' Equation
- Dispersion of Light
- Analysis of Prism
- Thin Prisms
- Some Natural Phenomena Due to Sunlight
- Defects of Lenses
- Optical Instruments
- Simple Microscope or a Reading Glass
- Compound Microscope
- Telescope
Electrostatics
- Concept of Electrostatics
- Electric Charge
- Basic Properties of Electric Charge
- Additive Nature of Charge
- Quantization of Charge
- Conservation of Charge
- Force between Charges
- Coulomb’s Law
- Scalar Form of Coulomb’s Law
- Relative Permittivity or Dielectric Constant
- Definition of Unit Charge from the Coulomb’s Law
- Coulomb's Law in Vector Form
- Principle of Superposition
- Electric Field
- Electric Field Intensity Due to a Point-Charge
- Practical Way of Calculating Electric Field
- Electric Lines of Force
- Electric Flux
- Gauss’s Law
- Electric Dipole
- Couple Acting on an Electric Dipole in a Uniform Electric Field
- Electric Intensity at a Point Due to an Electric Dipole
- Continuous Charge Distribution
Electric Current Through Conductors
- Concept of Electric Currents in Conductors
- Electric Current
- Flow of Current Through a Conductor
- Drift Speed
- Ohm's Law
- Limitations of Ohm’s Law
- Electrical Power
- Resistors
- Rheostat
- A combination of resistors in both series and parallel
- Specific Resistance
- Variation of Resistance with Temperature
- Electromotive Force (emf)
- Cells in Series
- Cells in Parallel
- Types of Cells
Magnetism
- Concept of Magnetism
- Magnetic Lines of Force
- The Bar Magnet
- Magnetic Field due to a Bar Magnet
- Magnetic Field Due to a Bar Magnet at an Arbitrary Point
- Gauss' Law of Magnetism
- The Earth’s Magnetism
Electromagnetic Waves and Communication System
- Foundations of Electromagnetic Theory
- EM Wave
- Sources of EM Waves
- Characteristics of EM Waves
- Electromagnetic Spectrum
- Radio Waves
- Microwaves
- Infrared waves
- Visible Light
- Ultraviolet rays
- X-rays
- Gamma Rays
- Propagation of EM Waves
- Ground (surface) Wave
- Space wave
- Sky wave propagation
- Communication System
- Elements of a Communication System
- Commonly Used Terms in Electronic Communication System
- Modulation
Semiconductors
- Concept of Semiconductors
- Electrical Conduction in Solids
- Band Theory of Solids
- Intrinsic Semiconductor
- Extrinsic Semiconductor
- n-type semiconductor
- p-type semiconductor
- Charge neutrality of extrinsic semiconductors
- p-n Junction
- A p-n Junction Diode
- Basics of Semiconductor Devices
- Applications of Semiconductors and P-n Junction Diode
- Thermistor
- Introdcution
- Definition: Centre of Mass
- Characteristics
- Real-Life Examples
Maharashtra State Board: Class 11
Introduction
The Centre of Mass (CM) is a fundamental concept in physics, especially when studying the motion of extended bodies. It is considered a hypothetical point where the entire mass of a body is assumed to be concentrated. This concept helps simplify the application of laws of motion to complex objects. Importantly, the centre of mass is a location, not a physical quantity itself. It serves as the particle equivalent of a given object for applying the laws of motion.
Maharashtra State Board: Class 11
Definition: Centre of mass
The centre of mass is a hypothetical point at which the entire mass of the body can be assumed to be concentrated.
Maharashtra State Board: Class 11
Characteristics
| Feature | Description |
|---|---|
| Location vs. Quantity | It is a location, and not a physical quantity. |
| Motion Application | It is the particle equivalent of a given object for applying the laws of motion. |
| Effect of Force | If a force is applied at this point, it causes only linear acceleration and not angular acceleration. |
| Rigid Body (Uniform Density) | For a rigid body of uniform density, it is located at the centroid. |
| Symmetric Body (Uniform Density) | For a symmetric rigid body of uniform density, it is located at the geometrical centre. |
| Change in Location | Its location can be changed only by an external unbalanced force. |
| Internal Forces | Internal forces (like during a collision or an explosion) never change the location of the centre of mass. |
| Mass Distribution | The position depends only upon the distribution of mass. |
| System of Particles | For a system of particles, the CM need not coincide with any of the particles. |
| Two Particles | For a system of only two particles, the CM divides the distance between them in an inverse ratio of their masses, meaning it is closer to the heavier mass. |
| Summation of Moments | It is a point about which the summation of moments of masses in the system is zero. |
| Axial Symmetry | If there is an axial symmetry, the CM lies on the axis of symmetry. |
| Multiple Axes of Symmetry | If there are multiple axes of symmetry, the CM is at their point of intersection. |
| Location Relative to Body | The CM need not be within the body. |
Maharashtra State Board: Class 11
Real-Life Examples
- Balancing: When balancing an object on a pivot, the line of action of the weight must pass through the centre of mass and the pivot.
- Sports (Fosbury Flop): A high jumper can momentarily pass their centre of mass below the bar (Picture 4.1) by bending their body, optimizing the jump.
- Objects with CM Outside: The centre of mass of a ring or a horseshoe lies in the open space, not within the physical material of the object itself.
Picture 4.1: Courtesy Wikipedia: Estimated center of mass/gravity of a high jumper doing a Fosbury Flop. Note that it is below the bar in this position. This is possible because our head and legs are much heavier than the fleshy part. The increase in the gravitational potential energy of the high jumper depends upon this point.
